Georg Cantor was a German mathematician who is partially credited with creating the foundation of modern general topology. Topology is a branch of mathematics that starts with logic and set theory, and applies measurement and classification such as ordering, shape, size, dimension. It is considered to be a foundational branch of mathematics in that many of the principles and theories have applications elsewhere. It is also the subject that made me want to know more, to keep studying mathematics. Georg Cantor and other mathematicians of his time had discovered theories that confounded logic as we know it, but were principally true nonetheless. One such theory deals with the question of “how many” and the set of all real numbers greater than 0 but less than 1.

Cantor Set - First Three Iterations

To begin with, start listing fractions beginning with 1/2 and continue to increase the denominator by 1 each fraction. The list goes like this: 1/2, 1/3, 1/4, 1/5, … and so on. Since there is no “largest denominator”, these fractions go on forever. There is an infinite number of them, and they are all contained between 0 and 1. Now the concept of infinity was not new in 1883. What indeed was new, was that not all infinities are equal. The above example is what mathematicians call a “countable” infinity. The definition of countable simply means that it has the same “quantity” as the integers themselves.

Now throw in irrational numbers as well. Take all of the numbers between 0 and 1. How many of these are there? Well, it was discovered that to even try to list them is impossible. In fact, take any two numbers in that interval. There will always be an infinite number between them… no matter how close together they are. Once a set of objects is uncountable, it begins to defy “common sense” logic. For example, it can be shown that there are just as many numbers between 0 and 1 as there are on the entire real number line. While this seems contradictory, it is the nature of microscopic infinity.

To put it in perspective, consider flying in an airplane high above the landscape. Start descending to the point where you can just make out a thin thread that is a highway stretching below. No individual cars can be seen; it appears to be just a thin line of light. As you descend, you begin to make out tiny shapes moving on the highway. The closer you get, the more possible it is to make out individual cars and discern spaces between them. As you approach, gaps between them increase and fewer are present in your field of vision. That is what the integers look like. If the cars represented all rational numbers, you would be curious to find that no matter how close you get, the row of cars appears unchanged. With each zoom, what before was invisible becomes visible; cars are in single file but just as dense and packed in. Still, you are seeing a countably infinite set. Even these are a mere drop compared to the ocean of the “reals”. The real number line has layers upon layers; they will not be placed in a line, for there are far too many of them. Endless microscopic rows appear, though moving along a thread.

Some had presumed that to be uncountably large, a set logically must have enough density so that its length (one-dimensional) or volume (multi-dimensional) would be greater than 0. But in 1883, Cantor showed this to be false. He presented the mathematical world with a set that had infinitely infinite number of members but whose length is 0. It was called the Cantor set. His set involved the number line, removing middle third of each remaining segment (see picture above) until all that remained was still uncountable but had total length 0. Here is a demonstration of a 2-dimensional example of a Cantor-type set:

Sierpinski Triangle

If this pattern were to be repeated an infinite number of times, it can be proven that the “black” portion that remains is an uncountable number of “dots” with area = 0. It is called the Sierpinski Triangle and is one of the most basic examples of what is called a fractal. Fractal Explained Hence the Cantor Set, infinitely complex but measure of 0, was born.

Come to think of it, House Majority Leader Eric Cantor and Georg Cantor have at least two things in common: both defy common sense and both arrive at seemingly contradictory conclusions. Perhaps there is a family connection.

As parents, we all have moods as well. There are days when I look at the kids’ choices of entertainment and think a box of Legos is all they need. Consider this, then. Without my poking and prodding, bribing and begging, my son has chosen to engage in activities on the electronic device that have helped him be able to count to 50, do simple addition, recognize all of the letters of the alphabet as well as write them, and put foods into basic food groups. I don’t have to give up my computer for this to happen, and waiting in the doctor’s office is 10,000 times less stressful. So… In anticipation of a busy Christmas shopping season, I did some research on available tablet technology dubbed “for kids”. Some are more “toy” than “tool” but I decided to cover them all here. Their function, price, and target age vary greatly. I also have taken the liberty to consider “total cost” that includes the device as well as games/activities/applications.

First to catch my eye is the VINCI Tab.

VINCI Tab

This tablet is marketed for babies through preschool, but at a pricetag of almost $400, I’d guess parents will not be looking at this rugged Android OS-based tablet for their toddler. With these specifications, it is robust enough to keep an 8-year-old busy. If mom or dad is technically savvy, any of the Android applications that do not require internet connection will work on this device. The appeal is that a child cannot access what the parent does not want him/her to access. It also is very rugged for kids. It connects to the parent’s Mac or PC via USB for software installation and for updates. This device is also the only TRUE touch screen Tablet made “just for kids” that I could find. The 8GB version with a 6 hour battery costs $479 while the 4 GB 3 hour one is $389. The upside of that is that many of the Android applications and games are free. The downside is that $400 still seems like a lot of money to put into a “toy”. The hardware is actually a Samsung Galaxy Tab with a few extras. If it were me, I’d look elsewhere.

Of the tablet-like options that require a stylus, the one that impresses me most is the Leapfrog LeapPad. The reason is that I am already familiar with the games and applications in the LeapFrog lineup. The educational value is not just lip service. I would put my son’s benefit (based on observation) at 60% learning, 40% fun and entertainment.

LeapFrog LeapPad

It is also more expensive than its main rival, but if one is looking for value, ROI (return on investment) then look no further. The interactive “games” teach new skills while children play, and this device is a real crowd-pleaser. On the features list, the makers of this device describe the screen as a “finger touch screen”. This is a bit misleading because it is not at all the same as the touch screen on your smartphone or iPad. One cannot make things larger, smaller, or shift objects around. The capability is limited to “selecting” something only. Even with the included stylus, images cannot be dragged or moved on the screen. To date, I have not found a single device designed specifially for child use that has this feature. The best part of this new device is that my son’s Explorer games will work on it. Win for current owners of the Leapster Explorer. 2GB built-in memory.

There are other LeapFrog products, including the Leapster Explorer bought for my son last Christmas. If one factors in the cost of the Explorer handheld + camera attachment, the $99 cost for the LeapPad with built-in camera is a bargain. As for applications, cartridge games cost about $25 each but there are also downloadable applications one can install for closer to $10.

Its nearest rival is the VTech InnoTab.

VTech InnoTab

At $89 it is $10 less than the LeapPad but it has no built-in camera. Both the LeapPad and the InnoTab require proprietary applications. This means that only VTech games and e-books will work on the InnoTab. The same is true for the LeapPad. The VTech InnoTab comes in two colors – blue (shown) and pink. What it does have: 64MB onboard memory and an SD card slot for videos/music. It supports music in MP3 format. This device also connects to a parent’s PC/Mac by way of a USB cable. According to device specs, the “touch screen” allows the user to select, drag, tap and flick. Again, this is limited functionality and will not at all be like an iPad screen.

Both of these options will “grow” with your child. They are great for preschoolers with a very kidlike appearance and appeal. But I have caught my 11 year old son enjoying his younger brother’s Explorer. Both of these have safety of not being wireless devices so that children will not accidentally activate applications with online capability. Unlike some handheld game systems, if the child is near another child with the same device, they cannot interact or engage in multi-player mode. This also is something many parents will find desirable.

All in all, the technology behind handheld learning systems for children has (for all practical purposes) arrived. Kids can use these without wishing they were using mom’s or dad’s laptop. And the price point is right.

http://www.technologyreview.com/blog/mimssbits/27049/?ref=rss

That is, the term “Web 2.0” will die. Now, to explain this we must understand what is meant by the term. Here is some of the history. In 2005, developers of search engines, like Google, came up with new ways to post, track, search, store, and tag data. This O’Reilly article explains it best: http://oreilly.com/web2/archive/what-is-web-20.html

Then, web technology grew to something more. In a Web 2.0 conference October, 2009, the evolving definition of Web 2.0 became more solid. http://www.web2summit.com/web2009/public/schedule/detail/10194

I think O’Reilly put it best in the following statement: “Web 2.0 is all about harnessing collective intelligence.” The social network, complete with hashtags and friends lists, is an expression of that collective intelligence. And it is largely done with text.

Now the way we experience the web will not change dramatically overnight. Many of us were moving to Web 2.0 when the general public was still getting a handle on Web 1.0. Most of the early changes were made without the average person even knowing it. What does this mean? One thing certainly is that if the creator of “Twitter” was hoping to get millions for its sale, those hopes just went out the window. Facebook and YouTube will be passé. So long to the boring teleconferencing via Illuminate! Bye bye Blackboard, bye bye. Well what could possibly be next, you might ask.

The answer may be in your children’s online gaming activity (or perhaps a spouse, or yourself, or the whole family). An idea that is still very much in its infancy, and is still rather like an expensive hobby, is a medium called “Second Life”. http://secondlife.com/whatis/ This has been around for more than 10 years, and there are still many details that need to be worked out before much of the framework goes mainstream. But it has a tarnished reputation since much of Second Life is restricted to those 18 years of age and older. It has a dark side due to a policy of uncensored smut. Serious business and/or educational/news entities never took it seriously, so Second Life remains pretty much the way it started, as an adult toy. It made an attempt to enter the teen market but failed. Therefore, it’s important to look at Second Life for its potential in technology, not necessarily what it is.

To see its potential, we are close to being able to reinvent any web presence into a visual, interactive, 3D screen on your computer monitor. It would be like participating in a LIVE an animated movie, and YOU represent one of the characters in that movie. Other characters are represented by actual people as well. There are two tools people are using in interesting ways. First is the design of a 3D world (what it looks like and placement of interactive objects). One is the Unity 3D engine. http://unity3d.com/unity/ Others are Jibe (works specifically with ReactionGrid) and Alice http://www.alice.org/index.php?page=what_is_alice/what_is_alice

In addition to the creation of a 3D world (the design of it), one needs a fairly robust computer on which the simulation environment is run. This is called hosting. Most people cannot maintain their own server, let alone set up the hosting. Both OpenSimulator http://opensimulator.org/wiki/Main_Page and Open Wonderland http://openwonderland.org/ offer this software free. Both are what is called “open source” which means development of the software is an openly collaborative project. Companies like Reaction Grid will do the design and hosting for you, but with a rather hefty pricetag. The state of Pennsylvania has its own hosting solution offered to K-12 institutions. It is called New Worlds, and can work with items/avatars/worlds previously created in Second Life. http://newworlds.cciu.org/ An interesting article that explains all this more thoroughly can be found here: http://thejournal.com/Articles/2011/01/11/Next-Stop-Open-Sim.aspx?Page=1.

We are already seeing the internet being used more and more in everyday activities. The state of Florida recently passed the “Digital Learning Now” Act which expands access to and use of online instruction. http://thejournal.com/articles/2011/07/26/a-closer-look-at-virtual-learning-models.aspx. The ways in which we think about education are changing. Educational institutions are generally the first to innovate, to bring new ways of thinking to the table. Then, other institutions will follow.

So what does this all mean? What is my take? We are about 18 months to 2 years away from working out the kinks and getting all the technology in place to make it happen, but the internet as you know it may look more like the Hypergrid of OpenSimulator. Consider what a typical school website looks like today. The main page contains pictures, graphics, links to other information on the school, and perhaps offers the ability for the user to download documents/forms. But most of it is in the form of text on a page. Clicking on the text moves the user to another page of information, and so on. The user had to type an address into Internet Explorer (or another browser) to view the school’s website. Suppose you have a son who plays soccer and attends the school. In order to schedule an appointment with the coach, you’d go to the website to find out the number to call, and dial. If the secretary is busy, you will have to call back. You have no idea if she is even in today. But it’s worth a try. Alas, no answer. You leave a message and hope she’ll return your call.

On a Hypergrid, the user would open his or her viewer software. Logging in would bring up the user’s “home” grid – an image of a 3D animated picture appears on the screen, complete with scenery, structures, a home, and rooms within the home, and furniture. The home décor is fully customizable, and as the user presses certain keys on the keyboard, he or she seems to be looking around. This is called a “first person” environment. The keys on the keyboard act as commands controlling a puppet on a stage. In this case, the stage is the image on the screen, and the puppet is a virtual character (called an avatar) representing the user (a human being, you) on that stage. Many of the images you see are representations of actual image files, documents, or actions taken with a computer.

As with the internet, the user can make himself or herself invisible to others (private). If one chooses to be public, his or her avatar will be visible to anyone else on that grid (website). One’s home grid is also private. The owner can control who can visit, or get access to their home. People who have home networks can see the usefulness of a 3D virtual home. So you want to listen to music from your digital library? No problem, just open the viewer software, use the keyboard to move the puppet (avatar) to the appropriate room of the house where the music library is, make your music selection, and press ‘play’. Obviously the digital music library on your computer is tied to this virtualized representation.

Now the user can choose to leave home and visit another grid (an outside web page using “old” lingo). To do this, we’re going to borrow an idea from a video game called Portal. Typing certain key combinations on your keyboard causes the appearance of a hole in the floor/ground of the scene in front of you. Type the site you want to go to, and your avatar jumps through the hole (portal) and is whisked away to another 3D scene. In OpenSimulator, your avatar merely disappears from one scene and reappears in the new one. Suppose the site name that you typed was the name of the school from the previous paragraph. On the screen in front of you, your avatar is positioned as though standing perhaps on a walkway leading up to the front door of the school, perhaps in the front entranceway near the Main Office. Rather than clicking on text on a web page, you literally move your avatar around this virtual environment. You may see other avatars as well. Ones with a question mark over their heads indicate you can stop and interact with them. If both of you have a microphone and speakers, you can talk as though face to face. Moving your avatar into the office, you see a table with papers on it. With the proper command, you can closely examine these papers. You see there is a form you need? Fine, click the download button. Your avatar will stop closely examining the papers once the download is complete. There is someone in the office talking with the secretary. You decide to sit in a chair and wait. When the secretary is available, you stand and send her a text message: “I wish to make an appointment to speak with Coach Jones”. The avatar for the secretary has a text bubble above her head. She appears to recognize you. “Can I tell him the purpose of the meeting?” You give her a brief purpose. She then gives you a date and time. Here is the kicker (no pun intended). The actual soccer coach of the school just had his calendar updated because the virtual secretary is tied to the actual secretary’s computer; so Coach Jones is now expecting you at that day and time, perhaps in person… or, perhaps, it is a virtual meeting. If you so desire, you can take your avatar to the virtual school and tell the virtual secretary you are here for your appointment. Just like the real world, avatars can behave like people. The coach’s avatar appears before your avatar and beckons for you to enter his virtual office.

Meetings can take place via communications software and all arranged in advance. It is possible that the creators of Skype (or someone else) may find a way to embed the ability to make a call right into this virtual environment. You will have the school’s phone number already entered. In mere moments, you could be talking with the real Coach Jones while your avatars register facial expressions, gestures, and body language (via your integrated webcam).

After your meeting, you hear a familiar sound. Ah, it is your avatar’s virtual cell phone ringtone. On your screen, your avatar reaches into her pocket and pulls out what appears to be a cell phone. Your spouse is calling to tell you he has left work and is on his way home. You did not have to log into your twitter account or other social media to find that out. He merely sent you a text message from his real cell phone and it was directed to your real cell phone as well as your virtual one.

So how does all this work? Well it doesn’t, not yet. This is why the technology to bring it all together is still about 2 years out. But for the first time, a few months ago when a colleague mentioned OpenSimulator to me, and my contacts at Carnegie-Mellon University have gotten excited about Open Wonderland, this form of internet seems not so far away.

Several factors will determine the integration of this kind of experience. Businesses will have to pay someone to “virtualize” their organization/operation. Secondly, every server that hosts Hypergrids, or virtual worlds will need to be able to handle hundreds of 3D avatars running a Live simulation program. Certainly, at first there will be limits on the number of avatars (people) present at any time on a particular site. To solve the problem of crashing servers due to too many join requests, we may see Hypergrids forming around the world in cells. Just like the state of Pennsylvania with its New Worlds simulation. Content will be geared to the audience desired, and each Hypergrid owner will determine the rules for conduct and appearance, just like current social networking sites. So, a Hypergrid cell geared for K-12 learning, discovery and play may look entirely different from a corporate Hypergrid cell. Retailers will then need to make decisions about which cells (or communities) in which they want a presence. It would be almost like the real world where chain stores maintain “brick and mortar” structures in strategic locations around the country. Companies would pay “rent” to have their virtual store operate in a Hypergrid. And same as the real world, certain locations would give them more exposure, therefore higher rent. But I’ll leave the economics for others to work out.

There are dangers to over-simulation (yes, I spelled that correctly). One of these dangers is that we can find ourselves so involved in the virtual world that we forget how to behave in the real one. Another is the tendency to think of the simulation as JUST a game, and kids being kids, the worst of human behavior can manifest itself due to the perceived anonymity an avatar brings. But this is a topic for another post. For now, I’m just getting started in the learning process.